
The spectrum is continuous with respect to \(\lambda \)  and \(b\) . In particular, when we shrink \(\lambda \rightarrow 0\)  we find the spectrum of Euler equations detailed in (REF ). However, when we shrink \(b\rightarrow 0\)  we obtain in part the simply connected spectrum (REF ) . In other words,
\(\left\lbrace \begin{array}{l}\Omega _{\mathbf {m}}^{\pm }(\lambda ,b)\underset{\lambda \rightarrow 0}{\longrightarrow }\Omega _{\mathbf {m}}^{\pm }(b)\\\Omega _{\mathbf {m}}^{+}(\lambda ,b)\underset{b\rightarrow 0}{\longrightarrow }\Omega _{\mathbf {m}}(\lambda ).\end{array}\right.\) 
These asymptotics are obtained for sufficiently large values of \(\mathbf {m}.\)  For more details see Lemma REF .

The regularity of the boundary, which is \(C^{1+\alpha }\) , is far from being optimal. We expect it to be analytic and one may use the approach developed in [1]} and successfully implemented in [2]} for the generalized quasi-geostrophic equations in the doubly-connected case.


